1. Introduction

In this chapter, I present one of the research areas existing in finance called bond portfolio immunization (BPI). My goal is to make it known to medical researchers dealing with immunity (resistance) of human organisms to diseases. I feature not only basic notions, problems,

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and solutions occurring in BPI, but also selected mathematical concepts and tools which proved to be instrumental in developing BPI. I do believe that such information has a good chance to be useful in creation of immunity against particular diseases. Bond investors are called immunizers if, possessing C dollars today, they must achieve an investment goal of L dollars q years from now (a human organism or a particular human organ must achieve a certain level of health q years from now); here L is the future value of C at time q under the current interest rates. This investment goal must be accomplished by means of an appropriately selected bond portfolio, even despite unfavorable sudden change (shift) in interest rates (appearance of a disease), having in mind that the present and future prices of all bonds depend solely on interest rates.

2.1. What are bonds?

Each bond with a face value (par value) of F dollars is a financial instrument which generates to its buyer (holder) specified payments every 6 months (or every quarter, every year) in the form of coupons, plus par value paid only at the termination (maturity) of the bond. The face value represents the amount borrowed by the seller (issuer) of a bond from the bond buyer. The coupons represent a predetermined percentage, say 3%, of face value F; if F = \$10,000, then all coupons paid per year sum up to \$300. Each bond has its own life span (maturity) of n years (3, 5, 10, 20 years, etc.) A bond portfolio (BP), by its definition, is a collection of different bonds with various maturities. Thus, each BP generates a more complicated cash flow pattern than a single bond does. A cash flow generated by a BP consists of various size payments ci, 1 ≤ i ≤ m, (coupons and par values generated by all kinds of bonds forming that portfolio) at certain dates t1, t2, t3, …, tm from an interval ½ � t0; T , where t<sup>0</sup> is the date when BP was purchased, while T stands for the

Practical Finance Strategies in Immunization http://dx.doi.org/10.5772/intechopen.78719 97

The present and future value of each bond, and consequently each bond portfolio, depends solely on current interest rates s tð Þ, which in the simplest case are identical for all maturities t, that is, s tðÞ� s, t∈½ � t0; T . By the term structure of interest rates, one understands a schedule of spot interest rates s tð Þ which are estimated from the yields (returns) of all coupon-bearing bonds. It is well known that interest rates are shaped under various random market forces.

The standard immunization problem relies on a construction of such a bond portfolio with the present value of C dollars that the single liability to pay L dollars q years from now (L is the future value of C) by means of the cash flow generated by BP will be secured regardless of how adverse changes in interest rates will occur in a future. This nontrivial problem is automatically solved by each zero-coupon bond maturing at time q with par value of L dollars. Thus, using medical terminology, one may say that such a zero-coupon bond possesses an innate

Besides, an investor may already possess bonds and would like to buy additional ones so that the created, in this way, portfolio BP with the present value of C dollars would secure the payment of L dollars q years from now. Having built such a portfolio, the investor would immunize (hedge) their own investment against a loss of its value at time q. We assume that the new term structure will always be of the form s\*(t) = s(t) + a(t), where a(t) belongs to a

On the other hand, the general immunization problem relies on a construction of such a bond portfolio BP with the present value of C dollars that multiply liabilities to pay Li dollars at specified instances of time will be secured by means of the cash flow generated by BP regard-

Immunization as a concept dates back as far as to articles [1, 2]. However, not until work [3] of Fisher and Weil was the impact of interest shifts on the design of immunization strategies

(natural) immunity. Unfortunately, in practice, such zero-coupon bonds rarely exist.

highest maturity of all bonds tradable on a given debt market D.

2.2. Standard and general immunization problem

less of adverse changes/shifts a(t) of interest rates in a future.

certain class of shifts (diseases).

2.3. Beginnings of immunization

Although, as it will be demonstrated in Sections 3.1, 3.2, and 3.3, immunization against all shifts is never possible, there are many results giving sufficient, or necessary and sufficient, conditions for immunization against a certain classes of shifts (certain diseases). It is worth to know that in the financial immunization, there is no such thing as acquired immunity (immunity that develops in a human after exposure to a suitable agent) and active immunity (acquired through production of antibodies within the organism in response to the presence of antigens).

Such types of immunization might theoretically take place on a bond market only if a bond holder had the right to change the coupon payments, which is completely out of the question. In other words, the immunization in financial reality has features of passive immunity, being in fact a short-acting immunity. On the other hand, however, a BP manager can achieve the state of a BP being all the time immune against a specific class of shifts, provided the manager regularly (every week or so) performs (if necessary) subsequent adjustments of his/her BP according to their expertise in the area of immunization theory.

Theorem 1 (Section 2.4.2) as well as Theorems 3 and 4 allows one to look at immunization from a different perspective. They enable one to identify all shifts a(t) (diseases) of the term structure s(t) of interest rates against which BP is already (fully) immunized, that is, protected against loss of its value at time q. Finally, having identified immunized (immune) bond portfolios, the natural question arises how to find among them the best ones. This topic is dealt with in Section 4.

Below, I shall present (i) what are bonds and bond portfolios; (ii) what is meant by standard (and general) immunization problem; (iii) historical development of immunization theory; (iv) overview of some recent results; (v) the concept of a Hilbert space, and a base in a linear space; (vi) application of orthogonal polynomials to description of the class IMMU of all shifts (diseases) against which a given bond portfolio is immunized; (vii) triangular functions as a base for the linear space IMMU; and (viii) the crucial role of the notions of duration and convexity in choosing the "best" immunized (immune) bond portfolios.
